Study on representations of algebraic groups and finite groups, and applications

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1. We studied the coinvariant algebras of finite subgroups G of SL(3, C). Particularly for the simple groups G of order 60, 168, we investigated the fiber of Hilb^<|G|>(C^3) over the origin (joint work with Nakamura (Hokkaido U.)). Also we showed that for the subgroups of SL(3, R), the fibers are the finite union of P^1. As a consequence of these studies, we could also show that the relations of Molien series and representation graphs, which have been studied by Springer, McKay, etc, for finite subgroups of SL(2, C), can be generalized for finite subgroups of SL(n, C) using the Koszul complex. (Shinoda, Gomi)2. Shinoda studied the properties of Gauss sums for finite reductive groups applying the theory of Deligne-Lustzig. Particularly for the semisimple representaions of classical groups and for the unipotent representations of G_2(q), we had explicit formula for the corresponding Gauss sums (jointly with N. Saito (Sophia U.)).3. We also obtained the following results :(1) Tsuzuki studied the Shintani functions on real Lie group U(n, 1) obtaining an explicit formula of the Shintani functions and also some multiplicity-free theorem for the corresponding represntation.(2) Nakashima realized irreducible representaions of finite dimensional quantum algebras of type A at root of unity through the representations of quantum group U_ε of type A of non-restricted specialization.(3) Goto clarified the structure of Goodman-de la Harpe-Jones subfactors using combinatorial methods.(4) Koga realized Wakimoto representation for the affine Lie superalgebra of type A and obtained a charcter formula for the heighest weight, representations (jointly with K. Iohara (Kobe U.)).